Comparison of Some Algorithms for Edge Gyrokinetics

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Comparison of (Some) Algorithms for Edge
Gyrokinetics
Greg (G.W.) Hammett Luc (J. L.) Peterson
(PPPL)Gyrokinetic Turbulence Workshop, Wolfgang
Pauli Institute, 15-19 Sep. 2008
w3.pppl.gov/hammett
Acknowledgments P. Colella, R. Samtaney
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Desired Algorithm Properties for Edge
Gyrokinetics

• Large variation in density, large amplitude
  fluctuations, large rbanana/L, wide range of
  collisionalities No clear separation of scales,
  Not useful or necessary to separate FF0df,
  stick with full F formulation
• Want to ensure particle conservation exactly
  (small charge imbalances lead to large fields)
  such as with finite volume, finite element,
  spectral methods.
• (some finite-difference, point-based
  semi-Lagrangian, and delta f weighted-particle
  algorithms (see Idomura, JCP 07) dont conserve
  particles exactly).
• Want positivity preserved even with large density
  variations (e.g. blobs advecting through low
  density SOL) many traditional algorithms have
  Gibbs phenomena oscillations around steep
  gradients that lead to negative densities.
• Want robust algorithms in addition to
  converging to the right answer in the appropriate
  limit, it shouldnt be too bad in other regime.
• Want to minimize numerical dissipation (though no
  need to eliminate it completely, dissipation at
  small scales actually models physical effects.)

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Desired Algorithm Properties for Edge
Gyrokinetics (2)

• It is surprisingly hard to find good algorithms
  (accurate, efficient, robust, not too hard to
  implement) that satisfy all of these properties
• There has been a lot of work over the past 30
  years on improving algorithms to address these
  types of issues for various kinds of CFD
  applications, with continuing advances in the
  last decade
• General category of shock-capturing or
  high-resolution upwind finite-volume
  algorithms, developed primarily for compressible
  shock problems in Euler/Navier-Stokes
  (aeronautics and astrophysics applications, etc.)
  But applicable to a wide range of problems
  including weather simulations, and our problems
• FCT, MUSCL, TVD, PLM, PPM, ENO, WENO, CWENO, SSP,
  MP, DG,

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Simplest Fluid Advection Algorithms2cd Order
Centered 1st order upwind

• Discrete grid, f(zj,t) fj(t) Conservative
  differencing

Std 2cd order centered differencing (okay for
smooth regions, phase errors too large for
sharp-gradient regions, gives unphysical
oscillations)
1st order upwind (eliminates unphysical
oscillations, but too dissipative)
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Arakawa Finite Differencing

• Clever differencing formula for Poisson brackets
  (in JCP special issue on most famous
  algorithms)
• Arakawa finite differencing has discrete analogs
  of conservation of
• In 1-D, (df/dy0, dPhi/dyv), Arakawa reduces to
  2cd order centered finite differencing. Although
  it has these nice conservation properties for
  Hamiltonian systems, it does not insure fgt0, and
  has significant phase errors at moderate kdx
  that can cause spurious oscillations.

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3rd order SSP-RK used here. Looks better at
CFL0.5 with 2cd order single-step
time-space-coupled time advancement, (becomes
exact at CFL1), but for complex flows there will
be regions at many different values of
CFLvdt/dx, incl. CFLltlt1.
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(My incomplete understanding of) Historical
Development of Shock-Capturing Fluid Algorithms

• Initial ideas from physicists (Boris, van Leer)
  (applied) mathematicians Phil Collela, Ami
  Harten, Stan Osher, Chi-Wang Shu, Bjorn Enquist,
  Eitan Tadmor,
• earliest numerical viscosity, simple upwind von
  Neumann Richtmeyer (50), Courant, Isaacson,
  Rees (52), Rosenbluth.
• Godunov (59) generalized upwind to multiple
  eqs. w/ shocks (Riemann solver), theorem only
  1st order near discontinuities
  piecewise-constant reconstructions
• Two indep. breakthroughs (FCT, van Leer)
  nonlinear switches enhance diffusion only near
  discontinuities or under-resolved features
• FCT (Flux-Corrected Transport) (71-79), Boris,
  Book. Zalesak version (79)
• van Leer (72-79), MUSCL (Monotone
  Upstream-Centered Schemes for Conservation laws)
  piecewise linear interpolation with slope
  limiters to avoid overshoots (2cd order in smooth
  regions, but const. near extrema, clipping)
• TVD (Total Variation Diminishing) (variations of
  2cd order van Leer)
• Colella Woodward 84 PPM (Piecewise Parabolic
  Method) (4th order for smooth solns, except
  const. near extrema) Widely-used gold-standard.
• ENO/WENO (Weighted Essentially Non-Oscillatory,
  87/94-96) Elegant solution to long-standing
  Gibbs osc. problem, arbitrary order (3rd, 5th
  typical) (related to fitting with a Sobolev
  norm?) Local operations, parallelizes easier
  than splines

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Suresh, Huynh 97

• Main idea behind these algorithms detect
  discontinuities / under-resolved features, revert
  to lower-order polynomial in non-smooth regions,
  allow discontinuities (allowed for hyperbolic
  eqs.), introduce minimum necessary numerical
  diffusion in non-smooth regions to preserve (or
  encourage) monotonicity, positivity.
• Suresh-Huynh (97) relaxed previous limiters to
  allow higher order interpolations near smooth
  extrema, 5th order in smooth regions, essentially
  a more efficient way to implement WENO
• Colella-Sekora (08) alternate way to relax
  piecewise-constant assumption at extrema, 4th
  order in smooth regions (even order no
  numerical diffusion in smooth regions)
• Discontinuous Galerkin looks like another
  potentially interesting approach

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Central differencing to determine slopes can lead
to overshoots in reconstruction

• Just going to higher order doesnt help near
  sharp gradient regions (Gibbs phenomena)

Top Fig. From R.J. Leveque, Finite Volume
Methods for Hyperbolic Problems, Cambridge Univ.
Press (2002). 2cd Fig. From C.B. Laney,
Computational Gasdynamics, Cambridge Univ. Press
(1998).
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Simplest Fluid Advection Algorithms2cd Order
Centered 1st order upwind

• Discrete grid, f(zj,t) fj(t) Conservative
  differencing

Std 2cd order centered differencing (okay for
smooth regions, phase errors too large for
sharp-gradient regions, gives unphysical
oscillations)
1st order upwind (eliminates unphysical
oscillations, but too dissipative)
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Higher-order upwind Methods withclever
monotonicity-preserving slope limiters

• Reconstruct f(z) in each cell, extrapolate to
  bdys

Piecewise constant 1st order upwind
Van Leers (MC) limiter Monotonized
Central in smooth regions, sj1/2 sj-1/2,
and becomes 2cd order accurate (upwind biased)
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Central differencing to determine slopes can lead
to overshoots in reconstruction

• MC limiter gives much more robust result.

From R.J. Leveque, Finite Volume Methods for
Hyperbolic Problems, Cambridge Univ. Press (2002).
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Arakawa in 1-D is simple centered 2cd order
method and has large overshoots and poor
performance on steep gradient regions, but it has
no numerical dissipation from the spatial
differencing (though there is some from the 3rd
order Runge-Kutta time advance).
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2-D vortex merger test case

• Test case used by Naulin Nielsen 03. We agree
  with them that WENO3 is fairly dissipative.
• Initialize 2 Gaussian vortices.

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Vortex Merger Test
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1-D Slices of Vorticity Along x 5 (for vortex
merger test)
SuHu extended PPM are essentially
non-oscillatory, but rigorously
non-oscillatory, but can be combined with FCT to
enforce rigorous positivity if needed.
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Without enough viscosity, Arakawa has enstrophy
pileup at high k. (Though little effect on
wavelengths at this time.)
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Even w/out explicit viscosity, high-order upwind
methods provide dissipation near the grid scale,
make spectra more realistic. (non-optimal subgrid
model, misses shearing?)
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Summary

• Suresh-Huynh 97 (5th order) Colella-Sekora 08
  (4th order), or hybrid between the two, look like
  very good options preserve high-order accuracy
  in smooth regions (including extrema), while
  still being robust and preventing artificial
  overshoots, provides useful dissipation near the
  grid scale.
• (Discontinuous Galerkin also looks interesting)

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References

• R.J. Leveque, Finite Volume Methods for
  Hyperbolic Problems, Cambridge Univ. Press
  (2002).
• D. R. Durran, Numerical Methods for Wave
  Equations in Geophysical Fluid Dynamics
• Arakawa. J. Comput. Phys. 1, 119- (1966).
• Cockburn and Shu. J. Sci. Comput. 163, 173-261
  (2001). Discontinuous Galerkin.
• Colella and Sekora. J. Comput. Phys. 22715,
  7069-7076 (2008).
• Liu, Osher and Chan. J. Comput. Phys. 115,
  200-212 (1994).
• Martin and Colella. Private Communication (2008).
• Naulin and Nielsen. SIAM J. Sci. Comput. 251,
  104-126 (2003).
• C.-W. Shu. ICASE Technical Report 97-65 (1997).
  Tutorial on ENO/WENO.
• Suresh and Huynh. J. Comput. Phys. 136, 83-99
  (1997).
• Zalesak. J. Comput. Phys. 31, 335-362 (1979).
  Improved form of FCT.
• Zhou, Li and Shu. J. Sci. Comput. 162, 145-171
  (2001).
• W. Rider, A Very Brief History of Hydrodynamic
  Codes, Sandia talk, 2007,https//cfwebprod.sandi
  a.gov/cfdocs/CCIM/docs/Rider_CSRI_June27_2007.pdf
• "Introduction to "Flux-Corrected Transport I.
  SHASTA, A Fluid Algorithm That Works", S. T.
  Zalesak 1997, JCP 135, 170. Nice 2-page review of
  historical place of FCT algorithm, and an
  introduction to the original FCT article,
  reprinted in this special issue of JCP
  celebrating its 30th anniversary.
• "Review Article Upwind and High-Resolution
  Methods for Compressible Flow From Donor Cell to
  Residual-Distribution Schemes", Bram van Leer,
  Commun. Comput. Phys. (2006).